UNIVERSITÀ DEGLI STUDI DI PADOVA
Dipartimento di Fisica e Astronomia ”Galileo Galilei”
Master Degree in Physics
Final Dissertation
Flavour changing neutral currents and axions
Thesis supervisor Candidate
Prof. Stefano Rigolin Jacopo Nava
Academic Year 2018/2019
Contents
1 Why going beyond the Standard Model 9
1.1 Experimental evidences . . . . 9
1.2 Theoretical hints . . . . 10
1.3 Naturalness and the hierarchy problem . . . . 12
2 Flavour physics 15 2.1 Construction of the Standard Model Lagrangian . . . . 15
2.1.1 Interlude on neutrino physics . . . . 18
2.2 Flavour structure of the Standard Model . . . . 18
2.2.1 CKM matrix and CP violation . . . . 21
2.2.2 Unitarity of the CKM matrix . . . . 24
2.3 FCNC and GIM Mechanism in the SM . . . . 26
3 Effective field theory in flavour physics 31 3.1 Effective field theories . . . . 32
3.1.1 Operator product expansion in the Standard Model . . . 34
3.2 Computation of Wilson Coefficients . . . . 35
3.2.1 OPE and short distance QCD corrections . . . . 36
3.2.2 Calculation of the amplitude in the SM . . . . 37
3.2.3 Calculation of Q1 and Q2 in the EFT . . . . 42
3.2.4 Extraction of the Wilson Coefficients . . . . 45
4 Strong CP problem and axions 49 4.1 The strong CP problem . . . . 49
4.1.1 Peccei Quinn solution to strong CP problem . . . . 52
4.2 QCD Axion . . . . 53
4.2.1 Invisible axion models . . . . 54
5 ALPs Effective Field Theory 57 5.1 FCNC top quark decay in the SM . . . . 58
5.2 Bosonic ALP Lagrangian . . . . 62
5.3 ALP and FCNC processes . . . . 63
5.3.1 Computation of Γ K+→ π+a . . . 69
5.4 Phenomenological bounds on ALPs couplings . . . . 70
A Loop integrals in D dimensions 75
B Computation of diagram c in Fig. 3.3 77 C Computation of the divergent parts of diagrams in Fig. 5.1 79
Introduction
The Standard Model of strong and electroweak interactions is the current the- ory of particle physics and its theoretical predictions have been in astonishing agreement with experimental data so far.
However, it is widely believed that the Standard Model (SM) needs an ultravi- olet completion, namely an extension in order to describe physical interactions properly at high energies. The main reason suggesting an UV completion of the SM is the fact that we are still missing a quantum theory of gravity and the energy at which quantum gravity effects become manifest, namely the Planck scale (MP ∼ 1019 GeV), constitutes the ultimate UV cutoff for the SM.
Nevertheless, there are some theoretical issues, which induce us to believe that the scale at which new physics (NP) appears should be much lower. The most compelling one is the so called hierarchy problem: the Higgs boson is a light particle compared to the Planck scale, but regarding the SM as an effective field theory with MP as UV cutoff, no argument is able to justify naturally the huge gap between the electroweak and the fundamental mass scale.
Trusting the Naturalness principle, we would expect the existence of NP around the TeV scale, which we are probing these years through LHC experiments.
Furthermore, cosmological issues like Dark Matter lead us to think that the SM is presumably an incomplete theory and new elementary particles still have to be discovered.
Within the search of a more fundamental model which solves the problems pre- sented, we focus in this thesis on an hypothetical class of particles known as Axion-Like Particles (ALPs). These particles inherit their name from the QCD Axion, postulated in 1977 by Peccei and Quinn to naturally solve the strong CP problem, namely the absence of CP violation in QCD. ALPs are the Pseudo- Nambu-Goldstone bosons (pNGb) of spontaneously broken U(1) global symme- tries, which appear in many extensions of the SM.
They can also acquire a mass madue to nonperturbative effects and in this case they consistute a promising candidate to explain DM nature.
While the theoretical difference between the axion and a generic ALP is that the latter does not need to solve the strong CP problem, the practical one is that for ALPs the symmetry breaking scale fa and ma can be treated as independent
parameters, while the two are instead related for the QCD axion. This fact, combined with cosmological observations, ultimately constrains the Axion mass to be set in the range between 10−5 and 10−2 eV. We actually focus on generic ALPs and thus consider fa and ma to be free parameters.
A powerful tool to describe ALPs contributions to physical observables is pro- vided by the EFT approach. In this framework the SM Lagrangian is extended with non renormalizable operators. Even though the new Lagrangian is not renormalizable, it nevertheless provides definite predictions up to the UV cutoff fa and in this energy range a perturbative expansion is well defined.
Within the EFT approach, NP contributions are included in a set of coeffi- cients associated with higher dimensional operators, called Wilson coefficients.
This affords to set up a model-independent analysis where NP contributions are parametrized by the Wilson coefficients. Then, the expression of such coefficients may be derived in specific NP models.
Among the most sensitive observables to NP we focus on those related to flavour physics. In the SM all the fermions carrying the same quantum numbers with respect to the SM gauge group come with three different replicas, known as flavours. The term flavour physics refers to interactions that distinguish among generations and in the SM all the source of flavour interactions is encoded in the Yukawa couplings of fermions with the Higgs field.
In particular we can make the distinction between flavour changing charged cur- rents (FCCC) and neutral currents (FCNC): the former change the flavour of a fermion current altering its electric charge, while the latter instead conserve the electric charge. Although FCCC are allowed to occur in the SM at tree level, FCNC processes are possible only at the loop level and are furthermore suppressed by the GIM mechanism.
Consequently, the study of FCNC processes is a powerful tool to detect possi- ble NP effects and enables us to put strong bounds on Beyond Standard Model (BSM) theories.
In this work we explore ALPs contributions to FCNC processes, formulating them in a model-independent approach via an ALP effective lagrangian. Our goal is to constrain ALP parameter space studying heavy meson FCNC decays.
In fact, while astrophysics and cosmology impose severe constraints on ALP in- teractions in the sub-KeV mass range and TeV scale can be tested at LHC, the most efficient probes of ALPs couplings in the Mev-GeV region come from pre- cision experiments performed at the charm/bottom quark scale.
In particular we analyze ALP couplings with electroweak bosons and fermions, assuming a flavour blind coupling. The comparison with data considers first each coupling separately, then the ensamble in combination and the resulting interference pattern is worked out in detail.
This thesis is developed in the following way: in chapter 1 we discuss the issues of the SM and we explain why we need an UV completion of the SM and at what energy we suppose it to occur. In chapter 2 we present briefly the con-
struction of the SM and we discuss in detail its flavour structure. In chapter 3 we describe the EFT approach and we present in detail an application of this method in flavour physics, within the SM framework. In chapter 4 we present the strong CP problem, we show how the axion arises as a natural solution and we shortly review the main features of axion models. In chapter 5 we introduce the ALP effective lagrangian and we put phenomenological bounds on its Wilson coefficients, analyzing FCNC processes. Incidentally, we discuss an example of a FCNC transition in the SM and we highlight the technical differences of the computations between an effective theory and a renormalizable theory.
Chapter 1
Why going beyond the Standard Model
1.1 Experimental evidences
The Standard Model of particle physics is the current theory describing the in- teractions of fundamental particles and has been successfully tested over the last fifty years.
However, there are various phenomena which induce us to believe that the Stan- dard Model (SM) may not be our ultimate theory of particle physics. The majority of these issues comes from the comparison between the SM predictions with cosmological observations, the most remarkable of which are:
• Dark Matter: Today there is experimental evidence that the SM particles constitute just the 5% of the energy density of the Universe. About the 26% of the energy density should be given by Dark Matter (DM), namely matter made of electrically neutral particles not included in the SM and at most weakly interacting with the SM fields.
Therefore the SM would presumably need a completion that introduces other particles in order to take into account for the presence of DM. There are various hypotheses about DM nature, between them for example super- symmetric particles, sterile neutrinos and the axion like particles (ALPs).
• Dark Energy: The remaining 69% of the energy density budget of the Uni- verse consists of Dark Energy (DE), an unknown form of energy responsible for the acceleration of the expansion of the Universe.
The simplest physical explanation for DE is that of an intrinsic, funda- mental energy of space given by the presence of quantum fields. However, attempts to explain DE in terms of vacuum energy of SM fields lead to a mismatch of 120 orders of magnitude. If this interpretation of the nature
of DE is correct, then the SM needs to be extended with other particles, in order to match the experimental value of the DE energy density.
• Matter-Antimatter asymmetry: It is well established from cosmological observations that in the Universe there is exceedingly more matter than antimatter, this fact is known as baryon asymmetry. In order to produce such an asymmetry, some constraints, known as Sakharov conditions, need to be satisfied [34]. These conditions for the production of baryon asym- metry are B violation, C and CP violation and the presence of an out of equilibrium phase in the early Universe. While such conditions are all qualitatively satisfied by the SM, the presence of an out of equilibrium phase would require the Higgs boson mass to be less than 80 GeV, which is ruled out by experiment. In addition the CP violation amount provided by the SM is too small to account for the baryon asymmetry.
Extensions of the SM, like Grand Unified Theories (GUT) and Supersym- metry (SUSY), can solve this problem, for instance introducing new sources of CP violation.
• Neutrino masses: According to the SM, neutrinos are massless particles, however from neutrino oscillation experiments we know that they do indeed have a non vanishing mass.
Mass terms for the neutrinos can be added to the SM, but they require an extension of the SM particle content, namely the introduction of a right handed neutrino for a Dirac mass term or new heavy degrees of freedom for Majorana mass terms.
Even if we already know from the previous observations that the SM cannot be the ultimate theory of particle physics, since it needs to be extended, to date no experimental result is accepted as disproving the SM at the 5σ level, which is fixed to be the threshold of a New Physics (NP) discovery in particle physics.
However, there are some observables, like the muon g − 2 factor, which show significant discrepancy from the SM predicted value and more sophisticated ex- perimental tests are needed to shed light on the nature of these discrepancies.
1.2 Theoretical hints
Independently from the compelling experimental reasons, one can also theoreti- cally argue that the SM cannot be a theory valid up to arbitrarily high energies.
First of all the SM does not include gravitational interactions and moreover a consistent quantum theory of gravity is still missing.
Nevertheless, the energy scale at which quantum gravity effects eventually be- come manifest represents the ultimate UV cutoff for the SM. Even in the absence of any other kind of NP effects up to Planck scale (Mpl∼1019GeV), the SM needs at least an UV completion to include gravitation.
However, there are some theoretical hints which make reasonable to think that new physics exists at lower energy, perhaps detectable by current collider exper- iments.
• One indication of NP below the Planck scale comes from the evolution of the three gauge couplings under the renormalization group. In the SM they almost merge at 1014GeV, while their unification is accomplished exactly at ∼ 1016GeV in supersymmetric extensions of the SM. This means that at such energy scale there could be just one type of gauge interaction with a larger gauge group, containing the SM gauge group as a subgroup.
This is very appealing from a theoretical point of view and supports the idea of GUT at such energy scale.
• Another theoretical clue comes from the smallness of neutrino masses, which would require an unnatural small Yukawa coupling with the Higgs field and the introduction of an unobserved right handed neutrino.
Neutrino small masses may be more naturally provided by the following five dimensional operator, known as Weinberg operator, which is the only one compatible with SM symmetries:
L5 = y
Λ( ˜φ†L)TC( ˜φ†L) (1.1) where ˜φ is the charge conjugate of the Higgs field, L is the leptonic left- handed doublet, C is the charge conjugation operator, Λ is the UV scale which originates this effective operator and y is an O(1) coupling constant.
Upon the Electroweak Symmetry Breakdown (EWSB), this operator pro- duces the following mass for neutrino:
mν = yv2
Λ , (1.2)
with v = 246.2GeV the VEV of the Higgs field and mν ∼ 0.1eV the neutrino mass. Inverting this relation we can get an estimate for the scale Λ, which yields a value O(1015) GeV.
• The Higgs boson mass is linked to the EW symmetry breaking scale, namely v = 246.2 GeV.
However, one expects that large loop corrections would make the Higgs mass huge, comparable to the Planck scale, unless there is an incredible fine tuning cancellation between the radiative corrections and the bare mass.
Trusting the principle of Naturalness, this fact suggests that new physics should appear at much lower scale, O (TeV).
• It’s an experimental fact that QCD interactions do not manifest CP viola- tion, this fact is known as the strong CP problem. The term θQCDGaµνG˜µνa ,
which in principle appears in the SM Lagrangian and leads to CP viola- tion in the strong sector, has an extremely small coefficient, θQCD ≤ 10−10. The Peccei-Quinn mechanism can explain ”naturally” this value and it also implies the existence of a new scalar particle, the QCD Axion.
1.3 Naturalness and the hierarchy problem
As we have seen in the previous section, most of the theoretical problems affect- ing the SM refer to the concept of ”Naturalness”. Therefore we need to define precisely what is the meaning of the Naturalness of a theory.
The definition of Naturalness given by ’t Hooft is [26]:
At any energy scale µ, a physical parameter or set of physical parameters αi(µ) is allowed to be very small only if the replacement αi(µ) = 0would increase the symmetry of the system.
Apparently, this definition seems to come from nowhere, but it becomes clearer recalling that, if the classical action of a QFT has a certain symmetry, then this symmetry, if not anomalous, must be fulfilled by the quantum action as well. As a consequence, if the parameter α is zero, also its correction δα should vanish, in order to preserve the symmetry at the quantum level.
If on the other side the symmetry is slightly broken by the appearance of the parameter, then the parameter will receive quantum corrections proportional to itself (δα ∼ α), because the symmetry should be restored in the α = 0 limit. An example of the first case is provided by massless gauge bosons, whose mass is constrained to be zero by gauge invariance. The second case is instead represented by fermion masses in the SM: setting them to zero restores the chiral symmetry, which protects them against large loop corrections. In fact, fermion masses get corrections of the following form:
δmf ∼ α
4πmflog Λ
mf (1.3)
that remain acceptably small even if the theory has a UV cutoff at MP l ∼ 1019 GeV. Thus, as a consequence of the ’t Hooft Naturalness principle, fermion masses are protected from planckian corrections and so a small value for mf/MP l is natural.
Conversely, the scalar masses fail to break any symmetry of the action. This implies that considering a scalar field φ, loop corrections δm2φ to m2φ depend quadratically on the UV scale Λ, rather than logarithmically as in the case of fermion masses, making unnatural a physical mass mφ far away from the fundamental scale of the Planck Mass.
This is indeed the case of the SM Higgs boson. Regarding the SM as an EFT
φ λf
f
f
λf
φ φ
φ φ
λs
φ
Figure 1.1: Two of the contributions to the Higgs mass in the Standard Model at one loop order: the diagram on the left involves a fermion loop, while the second is a self- correction of the Higgs propagator.
with UV cutoff at Λ = Mpl, since gravity has to couple with the Higgs field, the Higgs boson mass will receive a correction δm2h ∼ Mpl2. One may define the amount of fine tuning between the Higgs mass and the radiative corrections as:
f ≡ m2h
δm2h ∼ m2h
Mpl2 ∼ 10−34 (1.4)
An high fine tuning is regarded as unnatural in the absence of a mechanism to justify it. While there could be an ambiguity on the naturalness of a generic fine tuning parameter f, the fine tuning computed in Eq.(1.4) is unambiguously unnatural.
Thus, there is no way in the SM to naturally protect the Higgs mass from re- ceiving these large corrections, which would arise also even if the bare mass of the Higgs were zero.
Thus the problem of the Higgs naturalness eventually turns into a hierarchy problem: is actually the SM valid up to the Planck scale? If this is the case, then the question becomes why the EW scale and the Planck scale are so far away from each other.
Viable solutions require either introducing some new symmetry that protects the Higgs mass, such as Supersymmetry, or stating that the UV cutoff of the SM as an effective field theory is much lower than the Planck scale.
For example, setting the fine tuning to be f ≤ 10−2, then the UV cutoff of the SM should be at O (TeV). This means that there would be some new physics residing in the ”desert” between 103 and 1019GeV, whose lower region up to 14 TeV is being explored in these years by LHC.
Chapter 2
Flavour physics
2.1 Construction of the Standard Model Lagrangian
In this section we briefly present the construction of the Standard Model of par- ticle physics, formulated in its original version in the 60’ by Glashow, Weinberg and Salam.
The SM is a renormalizable quantum field theory and resting on the experimental observations, we set up the following theoretical framework:
1. A gauge symmetry group GSM, bearing the gauge boson fields
GSM = SU (3)C⊗ SU (2)L⊗ U (1)Y, (2.1) where we refer to the first subgroup as Quantum Chromodynamics (QCD), while the latter constitute the electroweak sector of the SM.
2. In nature we observe 4 gauge bosons mediating electroweak forces: one of them, the photon, is massless, while Z and W±are massive instead. Hence SU (2)L gauge symmetry must be spontaneously broken, leaving us with a residual one-dimensional group corresponding to U(1)EM, so
GSM → H = SU (3)C⊗ U (1)EM. (2.2) A doublet of complex scalar fields, φ, denoted as Higgs field, is introduced to allow the spontaneous breaking of GSM into the residual symmetry group H.
3. We introduce 3 generations of fermions, each of them consisting of 5 dif- ferent representations of GSM:
QLi(3, 2)1
6 , uRi(3, 1)2
3 , dRi(3, 1)−1
6
LLi(1, 2)1 , eRi(1, 1)−1. (2.3)
In this notation, for example, QLiare left-handed quarks (the index i=1,2,3 runs over families), which transform under the fundamental representation of SU(3)C and SU(2)L and carry weak hypercharge Y = 16. We can decompose the two SU(2)L doublets into their components in this way:
QLi=uLi
dLi
LLi =νLi
eLi
. (2.4)
Finally we have the Higgs field, which is a scalar transforming in the (1, 2)1 representation of GSM. 2
Using the above considerations we can write down a renormalizable Lagrangian for the SM, which can be split into four sectors: gauge boson, Higgs, fermionic and Yukawian:
LSM =LB+LH +LF +LY (2.5) where
LB = −1
4GaµνGµν,a−1
4Wµνb Wµν,b−1
4BµνBµν, (2.6) LH = (Dµφ)†(Dµφ) − µ2φ†φ − λ(φ†φ)2, (2.7) LF =X
i
i ¯QLiDQ/ Li+ ¯LLiDL/ Li+ ¯uRiDu/ Ri+ ¯dRiDd/ Ri+ ¯eRiDe/ Ri
, (2.8)
−LY = ¯QLiYijdφdRj+ ¯QLiYijuφu˜ Rj+ ¯LLiYijeφeRj+ h.c., (2.9) where Gµν, Wµν, Bµν are the field strength tensors associated to gluon fields (a=1...8), weak gauge bosons (b=1,2,3) and hypercharge boson Bµ, respectively.
φ = iσ˜ 2φ∗is the charge conjugated of φ, µ2 and λ are real parameters associated to the Higgs potential ( µ2 < 0and λ > 0) and Yu,d,e are 3 × 3 complex matrices known as Yukawa matrices.
The action of the gauge covariant derivative Dµon matter fields is determined by the representation of GSM under which they transform and takes the following
form:
Dµφ = ∂µ+ igWbµσ2b + ig0Bµ12φ
DµQL= ∂µ+ igsTaGaµ+ igWbµσ2b + ig0Bµ16QL DµLL= ∂µ+ igWbµσ2b − ig0Bµ1
2LL DµuR= ∂µ+ igsTaGaµ+ ig0Bµ23uR DµdR= ∂µ+ igsTaGaµ− ig0Bµ1
6dR DµeR= ∂µ− ig0BµeR,
(2.10)
where Ta and σa are the generators of SU(3) and SU(2) respectively, in the fundamental representation.
LB is the gauge boson part of the Lagrangian, containing boson kinetic terms and their cubic and quartic self interactions; this part of the Lagrangian contains three free parameters: g, g0 and gs, which are the gauge couplings associated to
the three simple subgroups of GSM.
LH is the Higgs doublet Lagrangian, containing its kinetic term, the interactions with the gauge fields and its potential. In order for φ to acquire a non vanishing V EV (v), we need to require µ2 < 0, while λ has to be positive in order to have a bounded potential. The relation between v and the free parameters of LH is easily obtained minimizing the potential and reads
v = r
−µ2
2λ (2.11)
The mass of the Higgs boson is given in terms of v, which is experimentally known from the muon decay, and λ, which is instead a free parameter of the theory. At tree level the relation reads
mh =
√
2λv2 = (124.97 ± 0.24)GeV (2.12) LF contains fermion kinetic terms and their interactions with gauge fields, while LY contains the Yukawa interaction between fermions and the Higgs field. This interaction produces fermion masses after Electroweak Symmetry Breaking (EWSB).
Due to Higgs mechanism, the electroweak gauge bosons W1, W2, W3 and B mix to create the states which are physically observable. These physical states are:
Wµ±= W1µ∓ iW2µ
√2 , (2.13)
Aµ
Zµ
= cos θW sin θW
− sin θW cos θW
Bµ
W3µ
, (2.14)
with the Weinberg angle θW = arctangg0 ∼ 0.23. W± and Z are the weak interaction mediator bosons, while A is the electromagnetic field. The former are massive, with mass linked to the VEV of the Higgs field in this way at tree level:
MW = gv
2 = 80.4GeV (2.15)
MZ=
pg2+ g02v
2 = 91.2GeV. (2.16)
We can rewrite LF replacing the gauge fields Waµ and Bµwith W±and Z. One then obtains:
LF = [LKin] − g
√
2(Wµ+J−µ+ Wµ−J+µ) − eAµJemµ − g cos θW
ZµJZµ (2.17) where
J−µ= ¯νLγµeL+ ¯uLγµdL Jemµ = −¯eγµe +2
3uγ¯ µu −1 3
dγ¯ µd J3Lµ = ¯LLγµσ3
2 LL+ ¯QLγµσ3 2 QL
JZµ= J3Lµ − sin2θWJemµ .
(2.18)
We commonly refer to Jµ± as charged currents terms (CC) and to JµZ as neutral currents (NC) terms.
2.1.1 Interlude on neutrino physics
Today three species of active neutrinos, one for each fermion generation, are thought to exist: this fact has been determined experimentally at LEP from the measurement of the invisible decay width of Z boson.
Neutrinos are different from other fermions because of some peculiar properties:
they are purely left-handed, namely no neutrino of right chirality has ever been observed up to now, and they interact with other particles only through the weak interaction, since they don’t carry neither electric nor colour charge.
Therefore, their observation is based on processes with very tiny cross sections, so that sensitive detectors and sophisticated techniques are required in their experimental hunt.
Even though in the original SM proposed by Glashow, Salam and Weinberg they were supposed to be massless particles, we know from the phenomenon of neutrino oscillation that they indeed possess a mass: actually, in neutrino oscillation experiments we are sensitive only to squared mass differences, and our best estimates on the sum of their masses is based on a combination of cosmological probes, which set the limit of Pimνi < 120meV(95%C.L.).
The fact of being electrically neutral opens the possibility that neutrinos are Majorana fermions, namely that they coincide with their own antiparticle. For neutrinos a Majorana mass term is allowed, which has the property of violating any U(1) global symmetry, in particular lepton number.
The other possibility to give neutrino mass is through the Yukawa interaction, introducing a right-handed neutrino for each generation, inert under electroweak interaction, which would produce the canonical Dirac mass term.
For both kind of mass terms we would have to extend the original definition of the SM, since the Majorana mass term for the SM neutrino can be provided only by higher dimensional operators, implying the existence of high energy fields and the Dirac mass term requires the introduction of a right handed neutrino.
Up to now we do not have evidence to establish whether neutrinos are Dirac or Majorana particles, therefore in the following discussion, for simplicity, we will assume that their mass is generated by a Dirac mass term, keeping in mind that this is far from being the only possibility.
2.2 Flavour structure of the Standard Model
The term Flavour physics refers to interactions that distinguish among fermion flavours, where by flavour we mean one of the different generations of fermion fields carrying the same quantum numbers, namely belonging to the same repre- sentation of GSM. In the SM all the source of flavour interactions is due to the
Yukawa matrices Ye,u,d,ν, as they are the only part of LSM which distinguish between families. In the absence of Yukawa interaction or in the case where Yukawa matrices are proportional to the identity, besides the Gauge Symmetry, LSM, extended with right-handed neutrinos, possesses a global U(3)6 symmetry, which can be split into its quark and lepton subgroup:
GF lavour = U (3)3Q⊗ U (3)3L, (2.19) under which fermion fields mix between generations for each representation in this way:
QL→ VQQL LL→ VLLL uR→ VuuR dR→ VddR
eR→ VeeR, νR→ VννR
(2.20)
where V’s are U(3) matrices. When considering Ye,u,d,ν 6= {0,13}, GSM is now broken down to a smaller group, known as the global accidental symmetry group of the SM. We call this symmetry accidental since it’s not a fundamental symmetry of our theory like the Gauge Symmetry, which was imposed a priori.
GF lavour → U (1)B⊗ U (1)L. (2.21) U (1)B is associated to baryon number conservation, while U(1)Lis associated to lepton number conservation. 1 Let’s now see how we can deduce the number of physical parameters of the SM looking in particular at LY.
Going in the unitary gauge the Higgs doublet takes the simple form
φ =
"
0
h+v√ 2
#
, (2.22)
Since Ye,ν,u,dare 3×3 complex matrices in flavour space they contain 4×2×32 = 72 parameters. However, since in a local QFT physics shouldn’t depend on the base used in field space, we are free to make a transformation in the fermion field space to diagonalize Yukawa matrices. As they are complex matrices, they are diagonalized by a bi-unitary transformation, with real and positive eigenvalues.
From Eq (2.9) the Yukawian Lagrangian reads:
−LY = h + v
√2 ¯eiRYeijejL+ ¯νRiYνijνLj + ¯uiRYuijujL+ ¯diRYdijdjL + h.c. (2.23) We now diagonalize Yukawa matrices making these unitary transformations on fermion fields:
fL→ VffL0 fR→ UffR0 (f = e, ν, u, d) (2.24)
1In the original version of SM, with massless neutrinos, individual lepton family numbers are conserved too at tree level. Taking into account for the presence of neutrinos, one U(1) combination, namely B + L, is anomalous at one loop, so the only exactly conserved quantum number associated to the accidental symmetry group of the SM is B − L.
where primed indices are referred to fermion fields in the mass basis. We finally get this expression for LY:
−LY = h + v
√ 2
e¯0Ri(Ye)ijDiage0Lj+ ¯ν0Ri(Yν)ijDiagνLj0 + ¯u0Ri(Yu)ijDiagu0Lj+ ¯d0Ri(Yd)ijDiagd0Lj+h.c.
(2.25) where (Yf)Diag = Uf†YfVf are the Yukawa matrices in the mass basis, which are related to quark masses by:
mf k = (Yf)kkDiag v
√2, (2.26)
where the index k runs over fermion generations.
While diagonalizing Yukawa matrices, the change of basis obviously modifies LF
too. In particular NC and CC terms undergo these transformations:
N C f¯LγµfL→ ¯fLVf†γµVffL= ¯fLγµfL
CC u¯LγµdL→ ¯uLVu†γµVddL= ¯uLγµVCKMdL
¯
νLγµeL→ ¯νLVν†γµVeeL= ¯νLγµUP M N S† eL
(2.27)
The neutral currents remain unchanged, while charged currents are affected by the change of basis both in the quark and lepton sectors. In the quark sector the change of basis produces the Cabibbo-Kobayashi-Maskawa (CKM) unitary matrix, given by:
VCKM = Vu†Vd=
Vud Vus Vub
Vcd Vcs Vcb Vtd Vts Vtb
, (2.28)
responsible for quark mixing, while in the leptor sector the change of basis pro- duces the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) unitary matrix, which is responsible for neutrino oscillations and is given by:
UP M N S = Ve†Vν =
Ue1 Ue2 Ue3 Uµ1 Uµ2 Uµ3
Uτ 1 Uτ 2 Uτ 3
. (2.29)
Therefore we conclude that only CC interactions feel the change of basis, i.e. the misalignment between the mass basis and the flavour basis. As a consequence, only Flavour Changing Charged Currents (FCCC) are admitted in the SM at tree level, while Flavour Changing Neutral Currents (FCNC) can appear only at one- loop order, so they are suppressed with respect to the former. We will analyze FCNC processes in the next section, in particular the Glashow-Iliopoulos-Maiani mechanism(GIM), which explains the suppression of FCNC in the SM.
To stress the importance that the study of FCNC had in the development of the SM, we remind that in 1964, when only up, down and strange quarks were thought to exist, on the basis of the suppression of FCNC, the existence of a fourth quark, the charm, was predicted. Actually the charm quark would be discovered only 10 years later, in 1974.
2.2.1 CKM matrix and CP violation
We have seen that the CKM is a 3 × 3 unitary matrix in flavour space, which is responsible for flavour changing transitions in the quark sector. Let’s consider now the general N generations case, where VCKM is so of dimension N. A gen- eral N × N unitary matrix has N2 parameters. N(N − 1)/2 of these parameters may be taken as Euler angles which one introduces when dealing with rotations in a N dimensional space, while the remaining N(N + 1)/2 are complex phases.
We can ask ourselves whether these phases are all physical or not.
We know from quantum mechanics that the phase of a wavefunction is not a measurable quantity, namely a wavefunction ψ and exp(iφ)ψ, where φ is a real number, are physically equivalent. The situation is the same in QFT: what mat- ters is not the absolute phase but the relative phases of different fields. Therefore we should examine which phases in VCKM are observable and which are not.
The phases of the fields are arbitrary, so we can redefine them performing phase transformations in this way:
dLk → eiφkdLk k = d, s, b
uLj → eiφjuLj j = u, c, t (2.30) Under the above transformations, for any number of families, what happens is that:
Vkj → ei(φk−φj)Vkj (2.31) where j and k denote an up-kind and down-kind quark respectively. Considering the effect of this field redefinition on the other pieces of LSM, NC terms are manifestly invariant because they are flavour conserving; LY is affected because it connects L and R fields, however this can be remedied by rephasing any right- handed quark field with the same phase as corresponding left-handed one, such that LY remains unchanged. From the previous equation, we see that the number of phases which can be reabsorbed is given by the number of phase differences φj − φk. Since we have 2N of such phases, 2N − 1 of them will be unphysical.
Therefore, CKM has N2 − (2N − 1) = (N − 1)2 total parameters, of which (N − 1)(N − 2)/2 are complex phases. The same reasoning applies identically to the PMNS matrix, for the lepton sector. Thus, we have two physical phases for the 3 fermion generations case (one for each mixing matrix), which combined with the 12 Yukawa couplings and the 6 Euler angles of the CKM and PMNS matrix give the number of 20 independent parameters for LY. Adding the gauge coupling constant g, g’ and gs, the free parameters of the Higgs potential λ and µ2, and finally the QCD theta term, we are left with 26 free parameters for the SM, with Dirac mass term for the neutrino.2
On the other side, for the case of 2 families, the CKM matrix has one rotation
2If neutrinos are Majorana particles, than the PMNS would possess 3 physical phases and the SM would have 28 free parameters
angle and no phases. In this case the quark mixing matrix has the simple form:
V = cos θC sin θC
− sin θC cos θC
, (2.32)
where θC is called Cabibbo angle. The importance of the irremovable phase in CKM matrix resides on the fact that it allows CP violation in the electroweak sector of the SM. To see this we recall the CP transformations for vector and axial fermion bilinears ( V and A) and for the gauge fields (W):
V ψ¯1γµψ2
−−→ − ¯CP ψ2γµψ1
A ψ¯1γµγ5ψ2
−−→ − ¯CP ψ2γµγ5ψ1
W Wµ±−−→ −WCP ∓µ
(2.33)
Considering the quark part of the Charged Current Lagrangian in the mass basis, we have:
LqCC = − g
√
2 ¯uiLγµVijCKMdjLWµ++ ¯dLi
γµVij†CKMujLWµ−, (2.34) Applying CP on LqCC we thus obtain the following Lagrangian:
L0CCq = − g
√2
d¯jLγµVijCKMuiLW−µ+ ¯ujLγµVjiCKM ∗diLW+µ, (2.35) We note that CP operation interchanges the two terms of LqCC and L0CCq except for the fact that Vij and Vij∗ are not interchanged. Thus CP is a good symmetry of the electroweak sector only if there is a choice of phase convention where all the couplings are real.
CP would not necessarily be violated in the three generations SM: for instance if two quarks of the same charge had the same mass, one mixing angle and the phase could be removed from CKM mixing matrix.
This happens because in this case we’d have an extra symmetry in the model, i.e. unitary transformations in the space spanned by the hypotetical degenerate mass quarks. Taking for example b and s quarks to be mass degenerate, we could build an s0 quark proportional to the linear combination Vuss + Vubb, whereby the up quark would couple only to two down-type quarks (d and s’) and not to the orthogonal combination b0. Thus the element Vub0 of our new CKM matrix would be zero and by use of unitarity we could show that in this case:
V =
cos θ sin θ 0
− sin θ cos φ cos θ cos φ sin φ sin θ sin φ − cos θ sin φ cos φ
(2.36)
This matrix leads to a CP conserving theory because it’s real and obviously this argument holds for any pair of up-type or down-type quarks. So we have six
